Function spaces with bounded \(L^p\) means and their continuous functionals
type: Item
instance of: scholarly article
title: Function spaces with bounded \(L^p\) means and their continuous functionals
zbMATH Open document ID: 1474.46065
DOI: 10.1155/2014/609525
author: Massimo A. Picardello
published in: Abstract and Applied Analysis
publication date: 2019-02-14T00:00:00Z
review text: Summary: This paper studies typical Banach and complete seminormed spaces of locally summable functions and their continuous functionals. Such spaces were introduced long ago as a natural environment to study almost periodic functions (Besicovitch, 1932; Bohr and Fölner, 1944) and are defined by boundedness of suitable \(L^p\) means. The supremum of such means defines a norm (or a seminorm, in the case of the full Marcinkiewicz space) that makes the respective spaces complete. Part of this paper is a review of the topological vector space structure, inclusion relations, and convolution operators. Then we expand and improve the deep theory due to Lau of representation of continuous functional and extreme points of the unit balls, adapt these results to Stepanoff spaces, and present interesting examples of discontinuous functionals that depend only on asymptotic values.
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 7022714
Wikidata QID: Q59039916
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2014/609525
OpenAlex ID: W1997630523
cites work: Q4481511
cites work: On the Banach spaces of functions with bounded upper means
cites work: Q5771475
cites work: On some types of functional spaces. A contribution to the theory of almost periodic functions
cites work: Q5526462
cites work: The class of convolution operators on the Marcinkiewicz spaces
cites work: Convolution on spaces of locally summable functions
cites work: Approximate identities on some homogeneous Banach spaces
cites work: Banach convolution algebras of functions II
cites work: Q3796164
cites work: On Generalized Harmonic Analysis
cites work: Uniformly Convex Spaces
cites work: On the uniform convexity of \(L^p\) and \(l^p\)
cites work: Q4107488
cites work: Finitely Additive Measures
cites work: Structure in real Banach spaces. I and II
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description: scientific article; zbMATH DE number 7022714